See [1]} for the relation of this equation with polygonal billiards, and see the second author's paper [2]} for a survey of this topic and links with some results at the basis of the present paper. A further connection comes from the fact that polygonal billiards are related to real multiplication in certain Jacobians (see [3]}), which occurs also concerning the Theorem of Poncelet (recalled below) which is at the basis of the billiard-map law in elliptical billiards: see J. Wilson's paper [4]} for proofs of Poncelet's theorem coming from real multiplication in Jacobians. We have presently no general direct explanations for these links but we hope this will be explored in some detail in the future.

